Axisymmetric resonant disturbances in a homogeneous wave guide
The author investigates the initial boundary-value stability problem for small localized axisymmetric disturbances in a homogeneous elastic wave guide, with free upper surface and the lower surface being rigidly attached to a half-space. The problem is formally solved by Laplace transform in time and by Hankel transforms of zero- and first-order in space. The author considers small three-dimensional localized perturbations in a linearly elastic isotropic layer of finite thickness and infinite horizontal extension. The analysis shows that axisymmetric resonant responses, with the growth of at least \(\sqrt t\), are present in a rather wide set of physically relevant wave guides, although all axisymmetric normal modes in any homogeneous wave guide are neutrally stable. The temporally resonant frequencies with the response of \(\sqrt t\) coincide with high-order temporally resonant frequencies with the response of \(t^{3/4}\) found in two-dimensional case. The abundance of temporally resonant frequencies in every homogeneous wave guide in the two-dimensional case is the basis for hypothesis put in the paper, i.e., that certain earthquakes can be triggered by a chain of events at the origin of which was a resonant response of a plate in the earth crust to a localized oscillation with a temporally resonant frequency. Since vibrations of low amplitude are always present in regions of seismic activity, the above hypothetical mechanism of triggering of earthquakes throught resonant growth can be of practical importance.
scientific article; zbMATH DE number 1300271